"Volume of hyperbolic 3-manifolds"의 두 판 사이의 차이
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| − | <h5 style="line-height: 2em; margin: 0px;"> | + | <h5 style="line-height: 2em; margin: 0px;">volume of figure eight knot complement</h5> |
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* 2.02988321281930725<br><math>V=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots</math><br> where D is [http://pythagoras0.springnote.com/pages/4633853 Bloch-Wigner dilogarithm].<br> | * 2.02988321281930725<br><math>V=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots</math><br> where D is [http://pythagoras0.springnote.com/pages/4633853 Bloch-Wigner dilogarithm].<br> | ||
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# L[x_] := Im[PolyLog[2, x]] + 1/2 Log[Abs[x]] Arg[1 - x]<br> f[x_, y_] :=<br> L[x] + L[1 - x*y] + L[y] + L[(1 - y)/(1 - x*y)] + L[(1 - x)/(1 - x*y)]<br> Print["five term relation"]<br> Table[f[i, j], {i, 0.1, 0.9, 0.1}, {j, 0.1, 0.9, 0.1}] // TableForm<br> N[3 L[Exp[2 I*Pi/3]], 20]<br> N[2 L[Exp[I*Pi/3]], 20]<br> N[3 (L[Exp[2 I*Pi/3]] - L[Exp[4 I*Pi/3]])/2, 20]<br> N[Pi^2*L[Exp[2 I*Pi/3]]/(3 Sqrt[3]), 20]<br> | # L[x_] := Im[PolyLog[2, x]] + 1/2 Log[Abs[x]] Arg[1 - x]<br> f[x_, y_] :=<br> L[x] + L[1 - x*y] + L[y] + L[(1 - y)/(1 - x*y)] + L[(1 - x)/(1 - x*y)]<br> Print["five term relation"]<br> Table[f[i, j], {i, 0.1, 0.9, 0.1}, {j, 0.1, 0.9, 0.1}] // TableForm<br> N[3 L[Exp[2 I*Pi/3]], 20]<br> N[2 L[Exp[I*Pi/3]], 20]<br> N[3 (L[Exp[2 I*Pi/3]] - L[Exp[4 I*Pi/3]])/2, 20]<br> N[Pi^2*L[Exp[2 I*Pi/3]]/(3 Sqrt[3]), 20]<br> | ||
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2010년 3월 31일 (수) 15:28 판
introduction
- hyperbolic 3-manifold : figure 8 knot complement
volume of figure eight knot complement
- 2.02988321281930725
\(V=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots\)
where D is Bloch-Wigner dilogarithm. - what is \(\zeta_{\mathbb{Q}(\sqrt{-3})}(2)\)? numrically 1.285190955484149
- L[x_] := Im[PolyLog[2, x]] + 1/2 Log[Abs[x]] Arg[1 - x]
f[x_, y_] :=
L[x] + L[1 - x*y] + L[y] + L[(1 - y)/(1 - x*y)] + L[(1 - x)/(1 - x*y)]
Print["five term relation"]
Table[f[i, j], {i, 0.1, 0.9, 0.1}, {j, 0.1, 0.9, 0.1}] // TableForm
N[3 L[Exp[2 I*Pi/3]], 20]
N[2 L[Exp[I*Pi/3]], 20]
N[3 (L[Exp[2 I*Pi/3]] - L[Exp[4 I*Pi/3]])/2, 20]
N[Pi^2*L[Exp[2 I*Pi/3]]/(3 Sqrt[3]), 20]
Chern-Simons invariant
Jones polynomial
links
history
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- 2010년 books and articles
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
[[4909919|]]
articles
- Volumes of Hyperbolic Manifolds and Mixed Tate Motives
- Alexander Goncharov, 1999
- Hyperbolic manifolds and special values of Dedekind zeta-functions
- Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월
- Commensurability classes and volumes of hyperbolic 3-manifolds
- A. Borel, Ann. Sc. Norm. Super. Pisa8, 1–33 (1981)
- 논문정리
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html[1]
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
- http://dx.doi.org/10.1007/s100529900935
question and answers(Math Overflow)
blogs
experts on the field